Clairaut's theorem

In $R^{n}$ , partial derivatives do commute. Also known as Schwarz rule.

Statement

Let $f (x, y)$ be a smooth function defined on an open set of $R^{2}$ , with continuous second partial derivatives. Then:

\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}

More generally, for $f : R^{n} \to R$ ,

\frac{\partial^{2} f}{\partial x^{i} \partial x^{j}} = \frac{\partial^{2} f}{\partial x^{j} \partial x^{i}}

provided the mixed partial derivatives are continuous.

Also known as

On a manifold with a linear connection:

\nabla_{μ} \nabla_{ν} f = \nabla_{ν} \nabla_{μ} f

because covariant derivatives of scalars reduce to partial derivatives, so they commute.

[\nabla_{μ}, \nabla_{ν}] T = R \cdot T

where $R$ is the Riemann curvature tensor. The curvature measures the failure of covariant derivatives to commute.